Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
History of randomness wikipedia , lookup
Indeterminism wikipedia , lookup
Probabilistic context-free grammar wikipedia , lookup
Dempster–Shafer theory wikipedia , lookup
Infinite monkey theorem wikipedia , lookup
Probability box wikipedia , lookup
Birthday problem wikipedia , lookup
Risk aversion (psychology) wikipedia , lookup
Boy or Girl paradox wikipedia , lookup
Inductive probability wikipedia , lookup
Chapter 4: Probability Rare Event Rule Forms the foundation for the inferential methods we will learn Ex: Consider tossing a fair coin. For an individual toss of the coin we are assuming that probability of heads = probability of tails = 0.5 Rare Event Rule for Inferential Statistics Give a particular assumption, if the probability of a particular observed event is extremely rare, we conclude that the assumption is probably not correct. Ex: Consider tossing a fair coin. What assumption are we making? If after 100 tosses we observe: TTTTHHTTHTHTTHHTHHTTHTTTTTTHH THTHHHHTTTTHHTTTHHTHHHTTTHHT THTTHTTTHHHHHTHHHTHTTHHHHHHT HHHTTTHHHHTTHTT What might we conclude? Rare Event Rule Basic Definitions Ex: Consider tossing a fair coin. For an individual toss of the coin we are assuming that probability of heads = probability of tails = 0.5 Toss a coin three times and record the sequences of heads (H) and tails (T) HHH HHT HTH THH If after 100 tosses of a different coin we observe: THHTTTTHHTTTTTTTTTTTTTTHTTHTH TTTTTHHHTTTTTHTTTTTTHTTTTTTTT TTTTTTTTHTTTTTHTTTTHTTTTHTTTT THTTTTHTTTHTT HTT THT TTH TTT Event: a collection of outcomes— all outcomes resulting in two heads and one tail S = sample space; The set of all possible outcomes from a procedure — all simple events What might we conclude? Simple event or sample point – one possible outcome that cannot be broken down into simpler components Notation Rule 1 • P denotes probability The Relative Frequency Interpretation of Probability • Capital letters such as A, B, C, … denote specific events • P(A) denotes the probability of event A occurring Define the probability of a specific outcome as the proportion of times it would occur over the long run (relative frequency of that particular outcome) Applies to situations that you can imagine repeating many times 1 Rule 1 Example Rule 2 number of times A occurred P A number of times the trial was repeated Classical Approach to Probability Define A to be the event “2 tails in a toss of three coins” • Assume that a procedure has n different simple events (n possible outcomes) We toss the three coins 25 times and observe 7 sets that have 2 tails P A ? 7 P A 0.28 or 28% 25 Rule 2 Example Consider the tossing three coins experiment Define A to be the event “2 tails in a toss of three coins” How many ways can A occur? HTT THT TTH • Each of the n outcomes are equally likely • If event A can occur in s of the n ways, then P A number of ways A can occur s number of different simple events n Law of Large Numbers As a procedure is repeated many, many times, the relative frequency probability (from Rule 1) of an event tends to approach the actual probability (Rule 2 for equally likely outcomes) How many different simple events are there? HHH HHT HTH THH HTT THT TTH TTT P A ? P A s 3 0.375 n 8 Assigning Probabilities Rule 3 • A value between 0 and 1 written either as a fraction or as a decimal fraction. • For the complete set of distinct possible outcomes of a random circumstance, the total of the assigned probabilities must equal 1. • The Personal Probability Interpretation • Personal probability of an event--the degree to which a given individual believes the event will happen--sometimes termed subjective probability • Restrictions on personal probabilities: Must fall between 0 and 1 (or between 0 and 100%) Must be coherent 2 Why Study Probability? • Want to be able to make inferences about a population from a sample or samples • Probability will allow inferences with a measure of reliability (or uncertainty) for the inferences • Initially, we will assume that the population is known and will calculate the probability of observing various samples from the population; i.e., use the population to infer the probable nature of the sample 3